Unsolved Problems

Let $r\geq 3$. A number $n$ is $r$-powerful if for every prime $p$ which divides $n$ we have $p^r\mid n$. Are there infinitely many integers which are...

Let $h(n)$ count the number of powerful (if $p\mid m$ then $p^2\mid m$) integers in $[n^2,(n+1)^2)$. Estimate $h(n)$. In particular is there some cons...

Let $A$ be the set of powerful numbers (if $p\mid n$ then $p^2\mid n$). Is it true that $ 1_A\ast 1_A(n)=n^{o(1)} $ for every $n$?", "difficulty":...

A critical vertex, edge, or set of edges, is one whose deletion lowers the chromatic number. Let $k\geq 4$ and $r\geq 1$. Must there exist a graph $G$...

Let $F(x)$ be the maximal $k$ such that there exist $n+1,\ldots,n+k\leq x$ with $\tau(n+1),\ldots,\tau(n+k)$ all distinct (where $\tau(m)$ counts the ...

Is there a function $f(n)$ and a $k$ such that in any $k$-colouring of the integers there exists a sequence $a_1<\cdots$ such that $a_n<f(n)$ for infi...

Let $S\subset \mathbb{R}$ be a set containing no solutions to $a+b=c$. Must there be a set $A\subseteq \mathbb{R}\backslash S$ of cardinality continuu...

Let $ f(n) = \sum_{p<n}\frac{1}{n-p}. $ Is it true that $ \liminf f(n)=1 $ and $ \limsup f(n)=\infty? $ Is it true that $f(n)=o(\log\log n)$ for all $...

Let $1<a_1<\cdots$ be a sequence of real numbers such that $ \left\lvert \prod_i a_i^{k_i}-\prod_j a_j^{\ell_j}\right\rvert \geq 1 $ for every distinc...

Let $A\subset \{ x\in \mathbb{R}^2 : \lvert x\rvert <r\}$ be a measurable set with no integer distances, that is, such that $\lvert a-b\rvert ot\in \...

Let $1=a_1<a_2<\cdots$ be the sequence of integers defined by $a_1=1$ and $a_{k+1}$ is the smallest integer $n$ for which the number of solutions to $...

Let $ s(n)=\sigma(n)-n=\sum_{\substack{d\mid n\\ d<n}}d $ be the sum of proper divisors function. If $A\subset \mathbb{N}$ has density $0$ then $s^{-1...

If $C,D\subseteq \mathbb{R}^2$ then the distance between $C$ and $D$ is defined by $ \delta(C,D)=\inf_{\substack{c\in C\\ d\in D}}\| c-d\|. $ Let $h(n...

Let $A\subset \mathbb{R}^2$ be a set of size $n$ and let $\{d_1,\ldots,d_k\}$ be the set of distinct distances determined by $A$. Let $f(d)$ be the nu...

Let $r,k\geq 2$ be fixed. Let $A\subset \mathbb{R}^2$ be a set of $n$ points with no $k$ points on a line. Determine the threshold $f_{r,k}(n)$ such t...

Let $f(k)$ be the minimal $n$ such that every set of $n$ consecutive integers $>k$ contains an integer divisible by a prime $>k$. Estimate $f(k)$....

Let $k(n)$ be the maximal $k$ such that there exists $m\leq n$ such that each of the integers $ m+1,\ldots,m+k $ are divisible by at least one prime $...

Let $f(n)$ be the maximal $k$ such that in any set $A\subset \mathbb{R}$ of size $n$ there is a subset $B\subseteq A$ of size $\lvert B\rvert\geq k$ w...

Let $u_n=p_n/n$, where $p_n$ is the $n$th prime. Does the set of $n$ such that $u_n<u_{n+1}$ have positive density?...

Let $Q(x)$ count the number of squarefree integers in $[1,x]$. Determine the order of magnitude in the error term in the asymptotic $ Q(x)=\frac{6}{\p...

Let $h(k)$ be Jacobsthal's function, defined to as the minimal $m$ such that, if $n$ has at most $k$ prime factors, then in any set of $m$ consecutive...

Let $p(a,d)$ be the least prime congruent to $a\pmod{d}$. Does there exist a constant $c>0$ such that, for all large $d$, $ p(a,d) > (1+c)\phi(d)\log ...

Does there exist a constant $C>1$ such that, for every $n\geq 2$, there exists a sequence $z_i\in \mathbb{C}$ with $z_1=1$ and $\lvert z_i\rvert \geq ...

Let $f\in \mathbb{Z}[x]$ be an irreducible non-constant polynomial such that $f(n)\geq 1$ for all large $n\in\mathbb{N}$. Does there exist a constant ...

Let $f\in \mathbb{Z}[x]$ be an irreducible polynomial of degree $d\geq 2$. Let $F_f(n)$ be maximal such that there exists $1\leq m\leq n$ with $f(m)$ ...

Let $f\in \mathbb{Z}[x]$ be an irreducible polynomial of degree $k>2$ (and suppose that $k eq 2^l$ for any $l\geq 1$) such that the leading coefficien...

Let $k\geq 2$, and let $f_k(n)$ count the number of solutions to $ n=p_1^k+\cdots+p_k^k, $ where the $p_i$ are prime numbers. Is it true that $\limsup...

Let $n\geq 2$ and $\pi(n)<k\leq n$. Let $f(k,n)$ be the smallest integer $r$ such that in any $A\subseteq \{1,\ldots,n\}$ of size $\lvert A\rvert=k$ t...

Is it true that, for every prime $p$, there is a prime $q<p$ which is a primitive root modulo $p$?...

For any fixed $k\geq 3$, $ R(k,n) \gg \frac{n^{k-1}}{(\log n)^c} $ for some constant $c=c(k)>0$....

Let $x_1,x_2,\ldots \in (0,1)$ be an infinite sequence and let $ A_k=\limsup_{n\to \infty}\left\lvert \sum_{j\leq n} e(kx_j)\right\rvert, $ where $e(x...

Let $f=a_0+\cdots+a_dx^d\in \mathbb{C}[x]$ be a polynomial. Is it true that, if $f$ has roots $z_1,\ldots,z_d$ with corresponding arguments $\theta_1,...

Let $x_1<x_2<\cdots$ be an infinite sequence of integers. Is it true that, for almost all $\alpha \in [0,1]$, the discrepancy $ D(N)=\max_{I\subseteq ...

Let $n_1<n_2<\cdots$ be a lacunary sequence of integers and $f\in L^2([0,1])$. Estimate the growth of, for almost all $\alpha$, $ \sum_{1\leq k\leq N}...

Let $n_1<n_2<\cdots$ be a lacunary sequence of integers, and let $f\in L^2([0,1])$. Let $f_n$ be the $n$th partial sum of the Fourier series of $f(x)$...

Call $x_1,x_2,\ldots \in (0,1)$ well-distributed if, for every $\epsilon>0$, if $k$ is sufficiently large then, for all $n>0$ and intervals $I\subsete...

For any $0<\alpha<1$, let $ f(\alpha,n)=\frac{1}{\log n}\sum_{1\leq k\leq n}(\tfrac{1}{2}-\{ \alpha k\}). $ Does $f(\alpha,n)$ have an asymptotic dist...

Are there infinitely many solutions to $\phi(n)=\phi(n+1)$, where $\phi$ is the Euler totient function?...

Let $c>0$. If $x$ is sufficiently large then does there exist $n\leq x$ such that the values of $\phi(n+k)$ are all distinct for $1\leq k\leq (\log x)...

Let $\frac{a_1}{b_1},\frac{a_2}{b_2},\ldots$ be the Farey fractions of order $n\geq 4$. Let $f(n)$ be the largest integer such that if $1\leq k<l\leq ...

Let $f_r(n)$ be minimal such that every graph on $n$ vertices with $\geq f_r(n)$ edges and chromatic number $\geq r$ contains a triangle. Determine $f...

Let $h_3(k)$ be the minimal $n$ such that there exists a triangle-free graph on $n$ vertices with chromatic number $k$. Find an asymptotic for $h_3(k)...

Let $R(k,l)$ be the Ramsey number, so the minimal $n$ such that every graph on at least $n$ vertices contains either a $K_k$ or an independent set on ...

Let $h(n)$ be minimal such that there is a graph on $n$ vertices with $n+h(n)$ edges which contains a cycle on $k$ vertices, for all $3\leq k\leq n$. ...

Let $f(n,k)$ be such that every graph on $n$ vertices and $k$ edges can be partitioned into at most $f(n,k)$ edge-disjoint complete graphs. Estimate $...

Is it true that, for every $k\geq 3$, there is a constant $c_k>0$ such that $ \mathrm{ex}(n,G_k) \ll n^{3/2-c_k}, $ where $G_k$ is the bipartite graph...

Is there a constant $c_t$, where $c_t\to \infty$ as $t\to \infty$, such that if $\mathcal{F}$ is a finite family of finite sets, all of size at least ...

If $R(k)$ is the Ramsey number for $K_k$, the minimal $n$ such that every $2$-colouring of the edges of $K_n$ contains a monochromatic copy of $K_k$, ...

If $R(k,l)$ is the Ramsey number then prove the existence of some $c>0$ such that $ \lim_k \frac{R(k+1,k)}{R(k,k)}> 1+c. $ ...

We say that a graph is $4$-chromatic critical if it has chromatic number $4$, and removing any edge decreases the chromatic number to $3$. Is there, f...